Entangled nematic phase

Chapter VI: Ultracold hypermetallic polar molecules for precision measure-

A.2 Entangled nematic phase

In this section we will focus on the nature of the nematic phase and its stability in the thermodynamic limit (TDL). The qualitatively mean-field nature of all other ordered phases permits a variety of straightforward arguments for their TDL stability.

The structure of the ground state nematic wavefunction in Fig. 5.3 suggests that itinerancy of Rydberg excitations plays an important role in stabilizing this state.

Since itinerancy of excitations and defects emerges perturbatively [150], we will rewrite the 2D Hamiltonian like,

ˆ

𝐻 =𝐻ˆ_𝐷 +𝜆𝐻ˆ_𝑄

= 1 2

∑︁

𝑖≠𝑗

𝑅⁶

𝑏

| ®𝑟_𝑖− ®𝑟_𝑗|⁶𝑛ˆ_𝑖𝑛ˆ_𝑗 −∑︁

𝑖

𝛿𝑛ˆ_𝑖 +𝜆

∑︁

𝑖

ˆ 𝜎^𝑥

𝑖.

Here the eigenstates of ˆ𝐻_𝐷are classical crystals, while𝜆≡ ^Ω

2 = ¹₂. To investigate the energy scales involved in the itinerant processes, we can begin by performing non-

Figure A.7: Energies of various low-energy classical crystals that are corrected up to 4th-order Rayleigh-Schrodinger perturbation theory (RSPT), which includes effects of single excitation itinerancy. Results are shown for (a) interactions up to nearest neighbor columns (|𝑥_𝑖 − 𝑥_𝑗| = 2 in the 2D lattice), and (b) full long- range interactions. Comparing to the 0th-order classical energies with long-range interactions (Fig. 5.3c), we see that the classically unfavorable states (due to longer- range terms) are stabilized by the perturbations. The energy scale of single-excitation itinerancy is therefore larger than the long-range terms.

Figure A.8: Example of a perturbative hopping process for a single excitation that emerges in the second-order Rayleigh-Schrodinger correction to a classical initial wavefunction (fourth order energy). In (a) a single excitation in an |𝑎 𝑏 𝑎 𝑏 𝑎 𝑏⟩-type state can hop while only violating a single Rydberg blockade constraint (denoted by a red line). In (b), a single excitation in an |𝑎 𝑏 𝑐𝑎 𝑏 𝑐⟩-type state must violate two blockade constraints when hopping. These quantum fluctuations generate entan- glement that preferentially stabilizes the|𝑎 𝑏 𝑎 𝑏 𝑎 𝑏⟩-type states, despite their higher classical and mean-field energy.

degenerate Rayleigh-Schrodinger perturbation theory (RSPT) starting from different initial column state crystals such as|𝑎 𝑏 𝑎 𝑏 𝑎 𝑏 . . .⟩, |𝑎 𝑏 𝑐𝑎 𝑏 𝑐 . . .⟩, etc (see Fig. 5.3).

Since the initial states that diagonalize ˆ𝐻_𝐷 are classical, with all sites having exactly

⟨𝑛ˆ_𝑖⟩ = 0 or 1, the first-order RSPT correction to the wavefunction (second-order energy) allows local superpositions of |0𝑖⟩ and |1𝑖⟩. This order of RSPT captures the energies of the mean-field ordered phases with high accuracy. The second- order RSPT correction to the wavefunction (4^th-order energy) allows for effective

“hopping” of a single excitation or defect from site 𝑖 to another site 𝑗. The first- and third- order corrections to the energy are zero when starting from a single column state crystal. In Fig. A.7 we compute the energies up to the 4th order RSPT correction for various starting states, using all long-range interactions as well as

interactions truncated to only be between neighboring columns (|𝑥_𝑖 −𝑥_𝑗| ≤ 2 on the 2D lattice; in this limit the classical energies of all low-energy column states are exactly degenerate). The states of type|𝑎 𝑏 𝑎 𝑏 𝑎 𝑏 ...⟩have the lowest RSPT energies in both interaction schemes. At the level of single-particle hopping, this is because hops within a single column from excited site 𝑦 to site 𝑦 ± 1 can be chosen to only violate 1 Rydberg blockade (𝑅_𝑏 ∼2.3) constraint (the new excitation is 2 sites away from a single excitation within its column, but still at least

√

5 away from all excitations in other columns). On the other hand, states with more|𝑎 𝑏 𝑐 ...⟩character have some single excitation hops which must violate two instances of the blockade (they can only hop to a position at 𝑦 ±1 which is 2 sites from an excitation in its own column and 2 sites from an excitation in the adjacent column). See Fig. A.8.

By comparing the results (Fig. A.7) with truncated interactions and full interactions, we can see that the energetic contributions of the quantum fluctuation induced itinerancy are larger than those of the long-range interactions, since the interaction favors |𝑎 𝑏 𝑐 . . .⟩states while itinerancy favors|𝑎 𝑏 𝑎 𝑏 . . .⟩states. These fluctuations create entanglement, because the in-column direction of low-energy hopping is highly dependent on the state of the adjacent columns. This suggests that the ground state should be populated by|𝑎 𝑏 𝑎 𝑏 . . .⟩states instead of the|𝑎 𝑏 𝑐𝑎 𝑏 𝑐 . . .⟩mean-field ground state, which is what we observe in the nematic ground state obtained from DMRG (see Fig. 5.3). This analysis only requires a low-order of perturbation theory to stabilize the nematic quantum crystal|𝑎 𝑏 𝑎 𝑏 𝑎 𝑏 . . .⟩, regardless of the system size, which provides clear evidence for its stability in the TDL.

However, if we take this non-degenerate RSPT style of analysis for each separate column state crystal to its extreme, we will notice that it diverges at high orders.

Specifically, when the perturbation series reaches an order comparable to the number of excitations in a single column, the itinerant perturbation term connects different column states such as |𝑎 𝑏 𝑎 𝑏 𝑎 𝑏 . . .⟩ and |𝑎 𝑐𝑎 𝑏 𝑎 𝑏 . . .⟩. These classical configura- tions are quasi-degenerate (exactly degenerate with truncated interactions) and the energy differences in the denominators go to zero. Unlike a typical cat state, where this degeneracy is a small constant, the present case accesses a manifold of exactly- and quasi-degenerate states that grows exponentially in the number of columns (see next subsection). This prevents a straightforward application of degenerate perturba- tion theory and hints at the possibility of a more exotic, non-perturbative entangled order.

In the results reported for the 12× 9 supercell in Fig. 5.3, we do in fact observe

non-trivial coupling between the exponentially large manifold of low-energy column states. It is clear the structure of the wavefunction (Fig. 3c) does not correspond to the single dressed |𝑎 𝑏 𝑎 𝑏 𝑎 𝑏 . . .⟩ state discussed earlier, since the wavefunction coefficients are distributed across the 6-fold permutations of the|𝑎 𝑏 𝑎 𝑏 . . .⟩configu- rations as well as the exponential manifold of other states around them. The structure of the entanglement spectrum between columns further supports the presence of a non-trivial (i.e. non- cat state), macroscopic entangled order (see next subsection) for the solution on the12×9supercell.

The stability of this order in the TDL is a much more complicated question than the single nematic quantum crystal discussed above. As we mentioned, a perturbative understanding will fail us. In a straightforward perturbative treatment (degenerate or non-degenerate) of the low-energy classical states, a size-extensive order of the series is needed to see coupling between the different quasi-degenerate states. For the finite supercells we are able to study, the energetic benefit of coupling the exponential classical manifold exceeds the finite cost to do so (due to finite effective column height). However, in the thermodynamic limit the coupling may go to zero due to the size-extensive order. The true TDL stability is not clear from this picture.

Numerically, a detailed finite-size scaling analysis is needed to resolve the fate of this macroscopically entangled order, but this is beyond the scope of our current calculations, and perhaps beyond current methods in general due to the small energy scales involved. Analytically, we note that the itinerant-crystalline nature of this order, along with its non-perturbative behavior, is similar to the incommensurate floating solid phase that emerges near the order-disorder transition in 1D Rydberg chains [150, 151]. Future efforts may seek to understand the nematic state through that lens. Nevertheless, it is undoubtedly clear from our analysis that fluctuations and entanglement are much larger in this nematic phase than in the other ordered phases.

D=3 low-energy projectors and the entanglement spectrum

The character of the nematic phase has been extensively discussed in terms of the classical configurations that make up the quantum wavefunction. It has been pointed out that all the low-energy (and thus the most relevant) classical configurations can be described in a succinct notation like |𝑎 𝑏 𝑐𝑎 𝑏 𝑐 ...⟩ in terms of compositions of 3 individual column states |𝑎⟩,|𝑏⟩, and |𝑐⟩ which are defined in Fig. 5.3. This notation is very suggestive of the idea that aqualitativemodel for the 2D state can be written as a 1D MPS with a local Hilbert space of dimension 3, spanning|𝑎⟩,|𝑏⟩,

Figure A.9: Structure of the entanglement spectrum for the 1D low-energy projector model of the nematic order, as described in Section A.2. Left: The entanglement structure when projectors are applied on every bond of the MPS, in an open-boundary conditions style. Right: Entanglement structure when an additional periodic pro- jector is applied between the first and last site in the MPS. This type of interaction is generally accounted for in the Γ-point basis calculations. Note the similarity of these Schmidt spectra to the nematic DMRG ground state shown in Fig. 5.3.

and|𝑐⟩.

A striking feature of the entanglement spectrum presented in Chapter 5 is the presence of 3 large eigenvalues, with a 1 : 2 degeneracy structure. To illustrate a possible origin of this pattern in the nematic phase, we create a simple model state with a similar entanglement spectrum.

The strength of the interactions where the nematic phase emerges (𝑅_𝑏 = 2.3) is such that configurations with adjacent columns in the same state (e.g. |𝑎 𝑏 𝑐𝑎 𝑏 𝑏 ...⟩) are much higher in energy than all configurations without any identical adjacent columns. The projector into the low energy subspace thus removes all configurations with adjacent columns in the same state. It can be written as a product of commuting

two-site operators,

P =Ö

𝑖

ˆ

𝑃_𝑖,𝑖+1, (A.6)

where ˆ𝑃_𝑖,𝑖+1=1− |𝑎 𝑎⟩ ⟨𝑎 𝑎| − |𝑏 𝑏⟩ ⟨𝑏 𝑏| − |𝑐𝑐⟩ ⟨𝑐𝑐|.

First, we will briefly comment on the eigenspectrum of P. It’s eigenvalues are all positive integers, with the smallest being 0. The number of 0 eigenvalues grows as∼ 2^𝐿, where 𝐿 is the length of the 1D chain. These eigenvalues correspond to all the possible arrangements of the individual column states that do not violate any constraints in Eq. (A.6). This reveals the origin of the exponential classical degeneracy that has been previously discussed in the 2D system.

Concerning the entanglement spectrum, we can apply the operator P to a simple product state|𝜓₀⟩(a𝐷 =1 MPS) containing an equal mixture of all possible column states i.e. |𝜓₀⟩=Î

𝑖(𝜆_𝑎|𝑎_𝑖⟩ +𝜆_𝑏|𝑏_𝑖⟩ +𝜆_𝑐|𝑐_𝑖⟩) where|𝑎_𝑖⟩, |𝑏_𝑖⟩, |𝑐_𝑖⟩represent states on column 𝑖. As long as |𝜆_𝑎| = |𝜆_𝑏| = |𝜆_𝑐|, then P |𝜓₀⟩ has the entanglement structure shown in Fig. A.9, which is very similar to that seen in the 2D nematic phase computed by DMRG.